He had first started thinking about the formula when doing his weight training. As the motto “no pain, no gain” implies, he thought, he could view weight training as a kind of trade-off between some level of current pain (or discomfort) and the future benefits that were supposedly going to accrue from his spiffy new body. Now, there are different types of pain and discomfort involved in any kind of physical training. Some people might say that just being on the treadmill for half an hour in and of itself constitutes discomfort, if for no other reason than being boring. If you run faster and start getting tired, or do some other exercise that requires a similar level of significant effort, that can get painful or uncomfortable. At some point you may run into real muscle pain and soreness, which comes closer to what people normally think of as pain.

With weight training, there’s the minor discomfort that comes from exerting yourself and sweating, but at the end of a set if you are pushing yourself there’s a very specific kind of muscle pain that the experts usually explain as the effect of a particular chemical being released in your muscles. No study has identified a single cause for this discomfort, although the fact that it occurs more quickly in a muscle with a limited blood supply suggests that the culprit is a product of muscle metabolism. Common wisdom is that this is lactic acid, although experts say that the pain may also be due to ionic shifts at the cell membrane level and actual changes in the muscle cell proteins themselves. In any case, it hurts! It’s pain!

And there are other types of pain associated with exercise as well. Your muscles can start getting painful or sore sometime after you finish your training (so-called “delayed onset muscle pain”), which Bob on occasion had experienced lasting for as long as three or four days. In fact, Bob had been having this specific type of pain in his triceps for the last two days after a particularly hard workout.

But all this pain is some kind of advance payment towards a well-formed, strong body, right?. Bob supposed that this kind of great body, once he got there, would probably bring some kind of future pleasure in and of itself (which he called “future intrinsic pleasure”)—first and foremost, the pleasure resulting from him being able to just sit there and think about how nice his body was or look at it in the mirror. Then there was the future pleasure Bob would derive from knowing that he had set a goal and achieved it! In addition, certainly walking down the street and having people sneak admiring looks at his muscular physique would bring yet more future pleasure; being able to talk to his friends and acquaintances about his working out, in a thinly disguised superior way, would yield additional pleasure. Probably being stronger and looking better would mean that Bob could have better sex, something associated with pleasure no matter who you talk to. It could also mean more and better girl friends, which could in turn easily lead to better sex.

There were additional types of pleasure as well, though, and that was what Bob was wrestling with in trying to put together his formula. Like any formula, Bob’s formula was supposed to yield a specific answer, in this case to a question that had been bothering him: what if he was working out so hard that the pain he was experiencing in the present actually outweighed all the future pleasure he could expect? That would call for taking it a little easier in his workouts. Conversely, what if with a little bit more exertion and pain, say 10% more, he could get maybe double the future pleasure? That would certainly be a worthwhile investment that he should choose to make. The formula he was working on should be able to capture all these relationships, and when solved for P, the pleasure/pain variable, could tell him exactly how much current pain it was worth enduring.

Things started getting tricky when Bob realized that the formula was going to include pain and pleasure happening at different points in time. But obviously pleasure right now and pleasure sometime in the future, or pleasure in two days and pleasure in two years, are not interchangeable. In a funny way, pleasure at any point in the future is unreal, Bob pondered, since it had not happened yet. On the other hand, he ruminated further, once you get into the future, well, pain that happened in the past is unreal as well. So how could the two be connected? Bob turned to finance for a possible answer. When calculating the value of a present and future flow of money, he knew, you could use the concept of discounted value; in other words, a dollar you get a year from now is worth maybe 95 cents now, assuming you could invest the 95 cents and with interest it would turn into a dollar in a year.

Bob applied this concept to pleasure, and named it DPF, or discounted pleasure flow. But there were problems with this. With money, you discount future inflows using something like the interest rate on treasury bonds. But what is the right discount rate on pleasure? Bob figured that it was higher than a typical interest rate, maybe as high as 90%, meaning that ten units of pleasure a year in the future shouldn’t be traded for much less than one unit of pleasure right now. But the more basic problem that the future pleasure is not happening right now, and in that sense is somehow worth nothing, remained. This conundrum led Bob to a major breakthrough in this thinking: the value of future pleasure is the pleasure that he could get right know thinking about that future pleasure. In other words, even if the future pleasure is in some sense just a chimera, the anticipation of that pleasure is something that he could experience right now. This led him to reinterpret the discount rate; it was really the coefficient that, when applied to a particular amount of future pleasure, gives the quantity of current pleasure that can be derived from sitting there and thinking about how great it’s going to be to have that much pleasure in the future.

At first blush, the pleasure discount rate seemed like it should be a constant, in the mathematical sense. But was it really? It certainly was something that might differ from person to person. In fact, it could be one of the fundamental aspects of a person’s personality. How much do YOU discount future happiness by? There may well be people out there who are working with a discount rate of only 50% per annum; they’ll trade one unit of pleasure now for a mere two units a year in the future. Ideally, then, the formula might also yield a solution to what the “right” discount rate was. Are you better off working with a discount rate of 90% or 50%? Bob’s intuition was that a lot of people were working with lower discount rates than his 90%; in other words, they were more aggressively investing in future pleasure than he was. But a single formula could not possibly be solved for two variables, the discount rate plus the pain versus pleasure ratio. It’s just basic math that solving for two variables requires two equations. But what was the second equation?

Then there was the problem that even for one single individual, the discount rate might, or maybe even should, change over time. Sort of like the physicists who recently discovered that maybe the speed of light was changing gradually. Things were getting more and more complicated.

But how did Bob get the idea for working on this formula at all? Bob still remembered that magic moment in high school math class when he used formulas to show how slicing horizontally through a cone with a plane results in an circle. He started with the formula for the cone, which is x2+y2=z2; then tossed in a formula for a horizontal plane, which is just z=1. Then a simple substitution gave him x2+y2=1, the well-known formula for a circle! If you tilt the plane a bit, you get an ellipse; and if the plane is oriented vertically, so that it sort of slices off part of the side of the cone, you get a

bola. That’s why circles and ellipses and

bolas are called “conic sections”—they are what you get when you slice a “section” of a cone with a plane. Anyway, even though this was pretty basic introductory calculus level stuff, Bob was quite impressed with himself; and his 11th grade math teacher, Mr. Vance, even singled him out in class to praise his seminal contribution. (Mr. Vance didn’t know that Bob had his eye on his willowy 10th grade daughter.)

The pleasure formula, if it could be built and solved like the formula for conic sections, held immense promise. Exercise was just one possible application of the formula. To take a really mundane example, most people say you should make your bed when you get up in the morning. It may not fall under the category of “pain” to make your bed, but it takes time and energy. But making your bed gives rise to future pleasure, although the “future” in this case is pretty much limited to the next 18 hours until you sleep in your bed again: the pleasure you derive from looking at the bed during the day every time you walk into your bedroom and thinking how nice it looks, the chance for the pleasure that would result from someone else who happens to peer into your bedroom and see your well-made bed and think what a well-organized guy you are, the pleasure stemming from the great feeling you have all day long, even if you are at work, knowing that your bed is sitting there in your bedroom all pretty and well-made, the pleasure that comes that evening from flipping back and slipping into the taut blankets and sheets. This may seem like a trivial example, but all these things are undeniably derived from the simple act of making the bed in the morning. But what is the real exertion/payoff ratio? Bob thought if it could be quantified, using his formula, which would once and for all answer the age-old question of should people really bother to make their beds in the morning?

Bob was still confused, though, by how the pain or discomfort or exertion and the happiness or pleasure could be quantified. Without quantifying them, there was no way to build a formula. In other words, what is one unit of pain? Or of happiness? Pain is a little bit easier. But still, Bob realized that care was required here. There is instantaneous pain, and then a total amount of pain resulting from a particular amount of instantaneous pain occurring over a period of time. He could use a lower-case p for instantaneous pain levels, and P for pain occurring over a time interval. p=0 is just sitting in your easy chair. Climbing up the stairs to his second-floor apartment was probably around p=0.1 for Bob. Since that takes about 10 seconds, P=1.0 for that whole process, where the units are pain-seconds. (This is a well-established concept; think of a car, which might have an instantaneous speed of 50mph; if it continues along this at this speed for an hour, it will have covered 50 miles.) The strong muscle pain that occurs towards the end of a weight lifting set is something like p=2.0; since that usually lasts about 10 seconds, we have p=20.0 for the set, or, if there are 12 such experiences in the course of a one-hour workout, p=240.0. If you assume that the pain level for the non-intense periods of the workout, which still include huffing and puffing and sweating, is twice that of climbing the stairs, or p=0.2, and lasts for 58 minutes, or 3480 seconds, we get a P of 696.0, for a total work-out P of 936.0. It may or may not be interesting that the more intense bursts of pain at the end of each set therefore account for around 25% of the total pain associated with the whole workout.

But then Bob had an odd thought. He must be thinking about the formula because he expected that figuring it out would increase his overall pleasure. After all, if he could determine the right level of momentary exertion or pain in relation to the amount of pleasure it would generate over all future moments of time, then he could adjust things to optimize the total amount of pleasure during his entire life. To take a simple example, maybe he could “pay” ten units of exertion now for a twenty units of discounted future pleasure flow. Or, alternatively, he could pay twenty units now to get forty in the future. Clearly, he should choose the latter, since it would net him an extra ten units of overall pleasure (twenty versus ten in the first case), even though it required more pain right now. But he couldn’t make this choice, indeed would not even know that he had it to make, unless he first figured out the formula, so in that sense figuring out the formula was worth those points. Then again, though, the whole process of mental effort in building and solving the formula was in itself a sort of pain, which in the worst case could actually be greater than the ten points of pleasure he could gain by successfully applying the formula once it was solved. That would mean that the whole formula project itself was perversely decreasing Bob’s overall lifetime pleasure total! But the problem was circular—he had no way of knowing whether or not it actually would until he finished working out the formula. In the worst case, he might work and work and work and never solve the puzzle, digging himself into an immense hole of pain and effort that he would never be able to climb out of.

But in a way Bob was gaining some pleasure each moment that he worked on the formula. This was distinct from the pleasure relating to his expectation for some future flow of pleasure. It was somehow intrinsic in the very process of working on the formula. But at the same time the exertion, which did not really extend to the point that most people would call painful, but which Bob still considered to fall into the broader abstract category that he was calling pain for the purposes of the formula, was quite real. And he had another feeling that could be called pain too, although maybe dissatisfaction would be a better word: the feeling that maybe, just maybe, the whole project was doomed to failure and was just a massive waste of time. There sure were getting to be a lot of elements in this equation! Exertion just working on the damn thing, pleasure from anticipating increases in future pleasure, pain from that uneasy feeling that it wasn’t going to work, and then the intrinsic pleasure from twiddling with the formula. (There was also undeniably another pain element in working on it, regardless of its chances for eventual success: a kind of low-level, droning pain, which came from the feeling that maybe he was missing the chance to work on other, more pleasure-inducing projects; this Bob called “opportunity loss pain”.)

But there was yet another fly in the ointment. It was indisputably true that if Bob did solve the formula then he would increase, probably greatly increase, his lifetime pleasure. (Besides the direct pleasure boost, he would also get lots of pleasure from becoming famous and respected as the man who had solved this problem, which come to think of it is almost certainly one that has been around for a long time with lots of different philosophers weighing in on it, although he was pretty sure that nobody had thought of the simple approach of approaching it through a formula.) This future pleasure gave him current pleasure as he pondered it. But let’s say that all his work on the formula turned out to be for nothing—in that case wasn’t it a contradiction that he could get current pleasure from anticipating future pleasure which wasn’t even certain to materialize? The right answer to this dilemma, Bob pondered, was to say that current anticipatory pleasure could be weighted based on the probability that the future pleasure will actually happen. For instance, if some future pleasure would give you 100 units of current anticipatory pleasure if it was absolutely sure to happen, then it would give you 25 units if there was a 25% chance that it would happen. Of course, Bob had no way of estimating the probability that he could actually finish working out the formula, so this insight was not of much help; but he thought his chances were reasonably close to 50/50.

The more Bob thought, the more complicating factors he came up with. Consider a person in an instantaneous state of happiness 10, for whatever reason; maybe he or she is having sex. This type of happiness is heavy on the intrinsic side, but there is also an anticipatory side—pleasure coming from the future prospects of having more sex like this, begin able to tell your friends about it, or getting more sex partners because of your improving technique—as well as a recall-based side, when the sex reminds you of all the good sex you’ve had in the past. The current and future and past pleasures are kind of all mixed together into a pleasure cocktail. But there’s another type of future pleasure involved here: the future pleasure resulting from looking back on this particular roll in the hay and thinking about how pleasurable it was. See the circularity? Bob was getting pleasure now, in part from anticipating pleasure in the future, where that pleasure in the future was in turn partly derived, through the act of recall, from the pleasure he was having now. In mathematical terms, the whole thing was in danger of breaking down, the pain or pleasure variables on both sides of the equation cancelling each other out, leaving nothing but a meaningless formula like 1=1, or, worse, 1=0. On the other hand, this was the sort of interdependency relationship which could be the key to the solution.

Bob thought that a good starting point for the formula might be:

Pl(t) = PI(t) + PA(t) + PR(t)

where Pl(t) is total pleasure at time t, PI is intrinsic pleasure, PA anticipatory pleasure, and PR recollection-based pleasure.

Now, assume simplistically that PA(t) is some amount of pleasure at dt time in the future, appropriately discounted using the anticipatory pleasure discount factor r1, whether you think that is 90% or 50%:

PA(t) = Pl(t+dt) * r1

and PR(t) is some past amount of pleasure also discounted, but this time into the future (assuming that more recent pleasures remain more vivid in your mind) by the recollection-based happiness discount factor r2:

Now he used a trick of mathematics based on the fact that the discount rates r1 and r2 and both less than 1, and thus when multiplied by themselves become asymptotically close to zero, to go ahead and get rid of those terms, yielding:

Pl(t) (1 – 2*r1*r2) = PI(t) + r1*PI(t+dt) + r2*PI(t-dt)

In plain English, what this says is that for purposes of calculating pleasure at time t, Bob needed to take into account intrinic happiness at time t+td, but not double future anticipatory happiness at time t+td based on overall happiness at time t+2td; that’s too far away and the multiple discount factors will conspire to make it irrelevant. By the same token, in calculating his happiness now, he didn’t need to take into account the portion of his future pleasure that was based on looking back and thinking how happy he was now; these two bounces back and forth in time end up multiplying two small discount factors by each other with the effect fading out quickly.

This equation can be solved, if values for r1 and r2 are assumed. For instance, if r1 and r2 are both assumed to be 0.1 (remember this means that a specific amount of future happiness is experienced anticipatorily as one-tenth that amount of happiness right now; and a specific amount of past happiness is also experienced in recollection as one-tenth that amount of happiness right now). The result is startling: given an intrinsic level of happiness of 10, our overall level of happiness is just under 12. According to this theory, Bob’s overall current happiness could best be maximized by first, of course, maximizing intrinsic momentary pleasure; but also, importantly, by decreasing his pleasure discount factors; in other words, experiencing past and future happiness as more happiness right now. This was obvious once you thought about it.

But actually, this is completely wrong, or rather, right only as far as it goes; it uses only a single point of time in the past and another single point of time in the future. In fact the anticipatory component of our current happiness, Bob realized, is the sum of all the discounted future happiness moments:

PA(t) = sum(T=t to death) [ r1^(T-t) * Pl(T) ]

and similarly for recollected pleasure:

PR(t) = sum(T=birth to t) [ r1^(t-T) * Pl(T) ]

But even without trying to solve these equations (Bob had the feeling that they probably weren’t solvable), just writing them down gave rise to a doubt in Bob’s mind. Wasn’t all this pleasure being recalled and anticipated just a little too much? In the worst case, it could totally overwhelm a guy. In fact, why wasn’t it overwhelming Bob all the time? The reason could be that people put up some kind of filter to keep out too much recalled and anticipated pleasure. Or maybe it’s less like a filter and more like a conscious decision to only go out and fetch some recalled or anticipated pleasure or pain when they need it, or want it. But there was a twist here too. Making this sort of conscious decision was a type of exertion, which Bob was classifying under the broad concept of pain. If he was going for a particular level of pleasure, and then decided to go out and grab some anticipatory pleasure in order to reach that level, he would have to grab a little bit more than otherwise required in order to make up for the exertion/pain involved in going out and grabbing the future pleasure in the first place. And wasn’t it possible that there was actually more potential intrinsic current happiness available than people actually experienced, and with a little bit of extra effort he could fetch more of that too?

Another problem. Bob had been thinking of pain as pleasure as lying along some numeric spectrum, with zero being the absence of either, negative values representing pain, and positive ones pleasure. But there were times when he was sure he felt pain and pleasure simultaneously. So the simplistic approach of saying that you could just subtract the pain number from the pleasure number and end up with an overall value somewhere on the pain/pleasure scale didn’t seem right. Bob did really think that the pain and pleasure that he sometimes felt didn’t combine or average out in his mind, but remained quite distinct. This would call for a major change in the formula. He would need two different variables, Pa for pain and Pl for pleasure; that in itself was simple enough. But he would also need a way to relate the two to each other—a sort of “price”, in pain dollars, for one unit of pleasure. He had assumed that 10 units of pain together with 20 units of discounted future pleasure would result in 10 units of lifetime pleasure, but if pain and pleasure were distinct, this would be like adding apples and oranges. It would be easy enough to define the pain and pleasure scales so that one unit of pain offset one unit of pleasure at a particular time for a particular person such as Bob, but couldn’t this easily be quite different for other people, or even change over time for the same person, or be subject to change through diligent effort? Perhaps people could do some kind of training or something that would have the effect of decreasing the cost of pleasure in pain dollars, allowing them to get more pleasure for less pain.

But even after you have pleasure prices denominated in pain, there’s an aspect that the whole monetary analogy doesn’t really deal with well. That’s the fact that the pain was there and is real. With money, you pay it and it is gone, but with pain, you pay it and it is still there and you have still undeniably experienced it. Pondering these complications, Bob was consumed with doubts about whether the whole formula project was in fact moving ahead at all.

The next thought hit Bob like a thunderbolt. What if pleasure and pain are not really smoothly changing over time, like a car’s speed, but instead are little tiny se

te bursts? Come to think of it, Bob had never really experienced pleasure or pain as something that was really continuous; they did seem to come in little spikes! That could also explain the observation that sometimes Bob seemed to feel pleasure and pain simultaneously but se

tely; maybe it was really a spike of pleasure and then a spike of pain happening immediately afterward! This also was a potential answer to the question of too much future and past pleasure and pain converging on the present. If the future and past pleasure and pain were really just spikes in between periods of nothing special happening, then even when added up over some period of time they would represent much less pleasure and pain to overwhelm you with in the present! This was sort of like a car that moved by spurting forward several meters, then coming to a complete stop, then spurting forward again.

This idea was quite hard to express mathematically, though. Bob vaguely remembered a mathematical concept of a function f(x) which was defined as

f(x) = ( 1, if x is rational (in other words, can be expressed as i/j) ( 0, if x is not

which seems to have the desired kind of spikiness; but it might not work because there were so many rational numbers that there would be way too many spikes. Besides, Bob was not sure how “pliable” this type of function was to further mathematical manipulation. Probably not very. Then Bob imagined something truly ridiculous: perhaps the mathematician who thought this up did so based on his own experience of the spikiness of his pleasure and pain, as a means of expressing it? It seems unlikely, but you never know.

The spikes seemed very interesting, but Bob couldn’t figure out where they were taking him. He was getting farther and farther away from getting any useful results from the formula project.

Perhaps more light would be shed on the matter by finding a case where people pay for current pleasure with current pain. The only example that came to Bob’s mind was that of the masochist, who whipped himself because it felt good; or partially hung himeself in order to get a better erection and orgasm. In the hanging case, since people sometimes die doing this, in addition to the direct pain of feeling strangled, there was probably also a conditional pain based on some small probability that you might die. Say that the pain of dying is 100, and the probability of dying is 1%, so the conditional pain associated with the possibility of dying is one unit; the pleasure value of the orgasm is ten, by definition; and perhaps the pain of the strangling feeling is four. That means a total of five units of pain for ten units of pleasure. So is it possible that the price of one unit of pleasure is one-half unit of pain? If you believe that the pleasure discount factor is 0.90, meaning 10 units of happiness a year from now is worth one unit of pain now, that meant that he should be willing to pay one unit of pain now for twenty units of pleasure a year in the future. But Bob didn’t really know where to go with this insight either.

In any case, Bob kept on working on the formula. It was just that, well, working on it gave him pleasure.